Boundedness of fractional heat semigroups generated by degenerate Schrödinger operators

Abstract

Let $L=-\frac{1}{\omega }\textrm{div}(A(x)\cdot \nabla )+V$ be a degenerate Schrödinger operator in ${\mathbb {R}}^{n}$, where $\omega$ is a weight of the Muckenhoupt class $A_{2}$, A(x) is a real and symmetric matrix depending on x and satisfies

$$\begin{aligned} C^{-1}\omega (x)|\xi |^{2} \le A(x)\xi _{i}\overline{\xi _{j}}\le C\omega (x)|\xi |^{2} \end{aligned}$$

for some positive constant C and all x, $\xi$ in ${\mathbb {R}}^{n}$, and V is a nonnegative potential belonging to a certain reverse Hölder class with respect to the measure $\omega (x)dx$. By the subordinative formula, various regularity estimates about the fractional heat semigroup $\{e^{-tL^{\alpha }}\}_{t>0}$ are investigated, where $L^{\alpha }$ denotes the fractional powers of L for $\alpha \in (0,1)$. As an application, we obtain the boundedness on the weighted Morrey spaces and BMO type spaces for some operator related to $L^{\alpha }$.